CSCI 246 Class 5 RATIONAL NUMBERS, QUOTIENT REMAINDER THEOREM

Transcription

1 CSCI 246 Class 5 RATIONAL NUMBERS, QUOTIENT REMAINDER THEOREM

2 Quiz Questions Lecture 8: Give the divisors of n when: n = 10 n = 0 Lecture 9: Say: 10 = 3*3 +1 What s the quotient q and the remainder r? Lecture 10: What notation would you use to say The floor of a b? What notation would you use to say The ceiling of a b?

3 Notes Quiz will be handed back tomorrow Will return grades next-day

4 Reminder What is Rational Numbers Q equal to?

5 Reminder What is Rational Numbers Q equal to? Q = a b a, b Z, b 0}

6 Reminder What is Rational Numbers Q equal to? Q = a b a, b Z, b 0 Which operations are rational numbers closed under?

7 Reminder What is Rational Numbers Q equal to? Q = a b a, b Z, b 0 Which operations are rational numbers closed under? Multiplication Addition

8 Divisibility

9 Divisibility Defn: if n, d Z d 0, then n is divisible iff k Z s. t. n = k d Consider 10 = 2 5 What s n? What s d? What s k?

10 Divisibility Defn: if n, d Z d 0, then n is divisible iff k Z s. t. n = k d What are the divisors of 21?

11 Prime Numbers: only divisible by 1 and itself

12 Prime Numbers: only divisible by 1 and itself Give the first 4 prime numbers

13 Transitivity of Divisibility Theorem: Let a, b, c εz

14 Transitivity of Divisibility Theorem: Let a, b, c εz Suppose a b and b c

15 Transitivity of Divisibility Theorem: Let a, b, c εz Suppose a b and b c Then?

16 Transitivity of Divisibility Theorem: Let a, b, c εz Suppose a b and b c Then a c Proof:

17 Transitivity of Divisibility Theorem: Let a, b, c εz Suppose a b and b c Then a c Proof: since a b there exists k 1 /*by definition of divisibility (b=k 1 *a) */ since b c there exists k 2 /*by definition of divisibility (c=k 2 *b) */ c = k 1 k 2 a

18 Lesson 9 Quotient Remainder Theorem Divisibility Defn: if n, d Z d 0, then n is divisible by d iff k Z s. t. n = k d Quotient Remainder Theorem for any integers a, b b 0, uniquely integers q, r s. t. a = q b + r, where 0 r b

19 Lesson 9 Quotient Remainder Theorem Divisibility Defn: if n, d Z d 0, then n is divisible by d iff k Z s. t. n = k d Quotient Remainder Theorem for any integers a, b b 0, uniquely integers q, r s. t. a = q b + r, where 0 r b What are the quotient q, and the remainder r in the following: 11 = 2* = 9*10 + 9

20 Lesson 9 Quotient Remainder Theorem Divisibility Defn: if n, d Z d 0, then n is divisible by d iff k Z s. t. n = k d Quotient Remainder Theorem for any integers a, b b 0, uniquely integers q, r s. t. a = q b + r, where 0 r b Mod: Quotient (reminder) q = a div b Remainder r = a mod d

21 Lesson 9 Quotient Remainder Theorem Divisibility Defn: if n, d Z d 0, then n is divisible by d iff k Z s. t. n = k d Quotient Remainder Theorem for any integers a, b b 0, uniquely integers q, r s. t. a = q b + r, where 0 r b Mod: Quotient (reminder) q = a div b Remainder r = a mod d Example: What is the quotient and remainder when 99 is divided by 7? q =?, a=?, d=?, r=?

22 Lesson 9 Quotient Remainder Theorem Divisibility Defn: if n, d Z d 0, then n is divisible by d iff k Z s. t. n = k d Quotient Remainder Theorem for any integers a, b b 0, uniquely integers q, r s. t. a = q b + r, where 0 r b Mod: Quotient (reminder) q = a div b Remainder r = a mod d Example: What is the quotient and remainder when 99 is divided by 7? q =?, a=?, d=?, r=? Rewrite the above in Modular arithmetic (mod) form:

23 Lesson 10 Floors, Ceiling functions Floor Function Assigns to the real number x the largest integer that is less than or equal to x Ceiling Function Assigns to the real number x the smallest integer that is greater than or equal to x

24 Lesson 10 Floors, Ceiling functions Floor Function Assigns to the real number x the largest integer that is less than or equal to x Ceiling Function Assigns to the real number x the smallest integer that is greater than or equal to x Examples: Floor of (1/2) =? Ceiling of (1/2) =?

25 Homework (Group) 1. Determine whether 3 7? Explain why or why not using the definitions? 2. What are the quotient when 101 is devised by 11? 3. What is 101 mod 11 equal to? 4. Let a = 3, b=9, c=81; use the proof outlined in the lecture video with these values to show that because a b and b c that a c 5. Show that if a b and b a, where a and be are integers, then a=b or a=-b 6. What is the floor of (-1/2) 7. What is the ceiling of (-1/2) 8. Prove or disprove that the ceiling of (x+y) = the (ceiling of x )+ (ceiling of y) for all real numbers x and y

26 Homework (Individual) 1. Determine whether 3 12? Explain why or why not using the definitions? 2. What are the quotient and remainder when -11 is divided by 3? 3. For the following, give the quotient and the remainder: a) 19 is divided by 7 b) -111 is divided by 11 c) 789 is divided by 23 d) 1001 is divided by What were the 3 cases given for the proof of the Quotient-Remainder Theorem? 5. Data stored on a computer disk or transmitted over a data network are represented as a string of bytes. Each byte is made up of 8 bits. How many bytes are required to encode 100 bits of data? (hint: report the celling or floor function of this problem which one makes sense here?)